Why would you even tell in the first places derivatives are simply fractions ? They’re not, unless in some very specific physical approximations and in that case don’t try to do anything funky, sticks with the basics stuff
At any given point p on an n-dimensional manifold, a 1-form defines an n-dimensional cotangent vector (in the language of bundles[3], a point in the fiber over p).
So how do we define fractions of sections or vectors?
In the article, Baez defines fractions of 2-forms on the plane as the pointwise ratio of coefficients of a basis vector, which he can do because, as he points out, the space of 2-forms at a point on a 2-dimensional manifold is a 1-dimensional vector space (more generally, for k-forms on an n-dimensional manifold, this dimension is n choose k, so only 1 for 0-forms [functions] and n-forms).
[1] https://mathworld.wolfram.com/BundleSection.html
There are many areas of mathematics that spun from this.